DOI.7/s66-3-7-5 The mean-fiel computation in a supermaret moel with server multiple vacations Quan-Lin Li Guirong Dai John C. S. Lui Yang Wang Receive: November / Accepte: 8 October 3 SpringerScienceBusinessMeiaNewYor3 Abstract While vacation processes are consiere to be orinary behavior for servers, the stuy of queueing networs with server vacations is limite, interesting, challenging. In this paper, we provie a unifie effective metho of functional analysis for the stuy of a supermaret moel with server multiple vacations. Firstly, we analyze a supermaret moel of N ientical servers with server multiple vacations, set up an infinite-imensional system of ifferential (or mean-fiel equations, which is satisfie by the expecte fraction vector, in terms of a technique of taile equations. Seconly, as N we use the operator semigroup to provie a mean-fiel limit for the sequence of Marov processes, which asymptotically approaches a single trajectory ientifie by the unique global solution to the infinite-imensional system of limiting ifferential equations. Thirly, we provie an effective algorithm for computing the fixe point of the infinite-imensional system of limiting ifferential equations, use the fixe point to give performance analysis of this supermaret moel, incluing the mean of stationary queue length in any server the expecte sojourn time that any arriving customer spens in this system. Finally, we use some numerical examples to analyze how the performance measures epen on some crucial factors of this supermaret moel. Note that the metho of this paper will be useful effective for performance analysis of Q.-L. Li (B G. Dai School of Economics Management Sciences, Yanshan University, Qinhuangao 664, People s Republic of China e-mail: liquanlin@mail.tsinghua.eu.cn J. C. S. Lui Department of Computer Science & Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong Y. Wang Institute of Networ Computing & Information Systems, Peing University, Beijing 87, People s Republic of China
complicate supermaret moels with respect to resource management in practical areas such as computer networs, manufacturing systems transportation networs. Keywors Supermaret moel Romize loa balancing Server vacation Join the shortest queue Expecte fraction vector Operator semigroup Mean-fiel limit Fixe point Performance analysis Introuction During the last three ecaes consierable attention has been pai to stuying queueing systems with server vacations. Queues with server vacations are always useful in moeling many real life situations such as igital communication, computer networs, prouction/inventory systems, transportation networs business systems. Various queueing moels with server vacations have been extensively reporte by a number of authors, for example, basic vacation policies inclue server multiple vacations, server single vacations, server woring vacations, N-policy, D- policy T-policy. Reaer may refer to Taagi (99, Dshalalow (995, 997 Tian Zhang (6 for more etails. In the stuy of queueing systems with server vacations, an important result is stochastic ecompositions of stationary queue length of stationary waiting time. For single-server queues, the stochastic ecompositions in the M/G/ queue with server vacations were first establishe by Fuhrmann Cooper (985; while in multiple-server queues, the conitional stochastic ecompositions for the M/M/c queues with server vacations were first analyze in Tian et al. (999. Up to now, extensive research on the single-server (or multiple-server queueing systems with server vacations has been well-ocumente, such as, by three survey papers of Doshi (986, 99 Alfa(3, by two boos of Taagi (99TianZhang(6. Until now, the available results of queueing networs with server vacations has been very limite. Note that the supermaret moels are an important class of queueing networs play a ey role in the area of networing resource management, thus the supermaret moel with server vacations is very interesting in the stuy of queueing networs with server vacations, it can also provie some new unersting valuable highlight for the orinary queues with server vacations which are escribe in Taagi (99 Tian Zhang(6. For queueing networs with server vacations, Vveensaya Suhov (5 firstiscussea supermaret moel with server On/Off vacations, analyze the stationary queue length istribution by means of the fixe point. However, the On/Off vacation iscipline is not accurate for unersting the vacation processes, because it is not clear why to begin a vacation how to en this vacation. This motivates us in this paper to further consier a supermaret moel with server multiple vacations, while for other cases such as server single vacations server woring vacations, we can similarly give performance analysis. Note that the results given in Vveensaya Suhov (5 is very interesting, it also inspires us to further provie an effective algorithm to compute the fixe point with respect to the choice number 3,which
have not been given a complete solution in the literature up to now. Note that the choice constant N, wheren is the number of servers in the supermaret moel. Dynamic romize loa balancing is often referre to as the supermaret moel. Recently, some supermaret moels have been analyze by means of queueing methos as well as Marov processes. For the simplest supermaret moel (that is, Poisson arrivals exponential service times, Vveensaya et al. (996 applie the operator semigroups of Marov processes to analyze the stationary istribution obtaine an important result: Super-exponential ecay tail. The super-exponential solution is a substantial improvement of system performance over that in the orinary M/M/ queue. At nearly the same time, Mitzenmacher (996 also analyze the same supermaret moel in terms of the ensity-epenent jump Marov processes,e.g., see Kurtz (98.Later, Turner (998 provie a martingale approach to further iscuss this supermaret moel. The path space evolution of the supermaret moel was stuie by Graham (a, b, 4 whoshowethat starting from inepenent initial states, as N the queues of the limiting process evolve inepenently. Lucza Norris (5 provie a strong approximation for the supermaret moel, Lucza McDiarmi (6, 7 showe that the length of the longest queue scales as (log log N/ log O(. The positive Harris recurrence of the Marov processes unerlying some supermaret moels was iscusse in Foss Chernova (998 Bramson(8,. Certain generalization of the supermaret moel has been explore in stuying various variations, for example, moeling more crucial factors by Mitzenmacher (999, Jacquet Vveensaya (998, Jacquet et al. (999 VveensayaSuhov (5; analyzing non-exponential server times or non-poisson input by Bramson et al. (,,, Vveensaya Suhov (997, Mitzenmacher et al. (, Li et al. (,, Li Lui ( Li(; fast Jacson networs by Martin Suhov (999, Martin ( SuhovVveensaya(. Up to now, there have been three excellent survey papers by Turner (996, Vveensaya Suhov (997 Mitzenmacheretal.(, one boo by Mitzenmacher Upfal (5. The mean-fiel equations mean-fiel limits play an important role in the stuy of supermaret moels. Reaers may refer to recent publications for the meanfiel moels, among which are Sznitman (989, Vveensaya Suhov (997, Le Bouec et al. (7, Benaim Le Bouec (8, Borenave et al. (9, Gast Gaujal (9,, Gast et al. ( TsitsilisXu(. This paper provies a clear picture for illustrating how to use mean-fiel moels to numerically analyze performance measures of complicate supermaret moels, is organize into three ey parts: (Part one setting up system of ifferential equations, see Section. (Part two theoretical support, see Sections 3 4. In Section 3, we use the operator semigroup to give some strict proofs for the mean-fiel limit (or propagation of chaos, which shows the asymptotic inepenence of queues in the supermaret moel with server vacations. Section 4 is a necessary supplementary part of the mean-fiel limit, in which the Lipschitzian conition is establishe for guaranteeing the existence uniqueness of solution to the system of limiting ifferential equations. (Part three performance analysis, Sections 5 6 provie a novel mean-fiel metho for being able to numerically analyze performance
measures of this supermaret moel after the basic preparation given in Sections 3 4. Although analysis of the supermaret moel with a finitely big N is very ifficult, we use the mean-fiel limit to be able to numerically analyze performance measures of one queue, the information of which will help us to unerst the total behavior of this supermaret moel as N.Itisworthwhiletonotethat some simulations in Bramson et al. (,, inicate that the asymptotic inepenence of queues can be forme well when N. Therefore, the metho of this paper is effective for performance analysis of complicate supermaret moels. The main contributions of this paper are threefol. The first one is to provie aunifieeffectivemethoforsettingupaninfinite-imensionalsystemof ifferential (or mean-fiel equations, which is satisfie by the expecte fraction vector in terms of a technique of taile equations. Specifically, we erive an important relation: the invariance of environment factor. Notethattheinvarianceof environment factor plays a ey role in our later stuy with respect to this supermaret moel. The secon contribution is the evelopment of a useful technique for establishing the Lipschitzian conition for the infinite-imensional fraction vector function f : R C (R for the general choice number. Note that the choice number was always assume in several important references, e.g., see Vveensaya Suhov (997, 5 Mitzenmacher et al. (. As seen in this paper, the case with has a special structure in the system of nonlinear equations satisfie by the fixe point, which is easily ealt with from some simple computation; while for the case with 3, this paper gives some new interesting results when establishing the the Lipschitzian conition, which leas to the strict proofs for the mean-fiel limit. The thir contribution of this paper is to provie an effective algorithm for computing the fixe point, also to provie performance analysis of this supermaret moel. Note that our algorithm has a ey which has the ability to etermine the bounary probabilities in the system of nonlinear equations satisfie by the fixe point. The remainer of this paper is organize as follows. In Section, weescribe a supermaret moel of N ientical servers with server multiple vacations, introuce the sequence of fraction vectors which express the supermaret moel as infinite-imensional Marov processes, set up an infinite-imensional system of ifferential equations satisfie by the expecte fraction vector in terms of a technique of taile equations. In Section 3, we use the operator semigroup to provie a mean-fiel limit for the sequence of Marov processes, which asymptotically approaches a single trajectory ientifie by the unique global solution to the infinite-imensional system of limiting ifferential equations. In Section 4, we provie a unifie effective metho for organizing the Lipschitzian conition for the infinite-imensional fraction vector function f : R C (R. Then we apply the Lipschitzian conition the Picar approximation to show that the limiting expecte fraction vector is the unique global solution to the system of limiting ifferential equations. In Section 5, weprovieaneffectivealgorithmtocompute the fixe point of the infinite-imensional system of limiting ifferential equations. In Section 6, we use the fixe point to give performance analysis of this supermaret moel, incluing the mean of the stationary queue length in any server the expecte sojourn time that any arriving customer spens in this system. Furthermore, we use some numerical examples to analyze how the performance measures epen
on some crucial factors of this supermaret moel. Some concluing remars are given in the final section. A supermaret moel with server multiple vacations In this section, we first escribe a supermaret moel of N ientical servers with server multiple vacations. Then we introuce the sequence of fraction vectors, which are use to express the supermaret moel as infinite-imensional Marov processes. Finally, we provie a unifie effective metho to set up an infiniteimensional system of ifferential equations satisfie by the expecte fraction vector of the supermaret moel in terms of a technique of taile equations. The supermaret moel consists of N ientical servers, where each server has an infinite buffer. The service times of each server are i.i.. with an exponential istribution of service rate µ. The vacation process of each server is base on the multiple vacation policy: When there is not any customer at one server its buffer, it immeiately taes a vacation eeps taing vacations until it fins at least one customer waiting in the server or its buffer at the vacation completion instant. The vacation time istribution of each server is exponential with vacation rate θ >. The common input flow is Poisson with arrival rate Nλ for λ >. Uponarrival,each customer chooses servers from the N servers inepenently uniformly at rom, joins the one whose queue length is the shortest. If there is a tie, servers with the shortest queue length are chosen romly. All customers in any server will be serve in the first-come-first-serve (FCFS manner, the arrival, service vacation processes are inepenent of each other. Figure provies a physical illustration for the supermaret moel of N ientical servers with server multiple vacations. Lemma The supermaret moel of N ientical servers with server multiple vacations is stable if < λ <. Fig. Asupermaretmoel with each customer choosing the loaing of servers Server µ, θ Possion input Nλ Server µ, θ Each customer probes servers Server 3 µ, θ Server N µ, θ
Proof If, thenthissupermaretmoelofn ientical servers with server multiple vacations is equivalent to a system of N inepenent M/M/ queues with server multiple vacations. From Chapter of Tian Zhang (6, it is seen that the M/M/ queue with server multiple vacations is stable if ρ λ/µ λ <.Usinga coupling metho, as given in Theorems 4 5 of Martin Suhov (999, it is easy to see that for a fixe number N,, 3,...,thissupermaretmoelofN ientical servers is stable if ρ λ <. This completes the proof.. An infinite-imensional Marov process For this supermaret moel, let L (N be the number of woring servers with at least customers (the serving customer is also taen into account at time t, M (N l the number of vacation servers with at least l customers at time t. We write U (N V (N l L(N N,, M(N l N, l. Clearly, U (N for V (N l for l are the fractions of these woring servers with at least customers at time t the fractions of these vacation servers with at least l customers at time t, respectively.set U (N (U (N, U (N, U (N 3,... V (N (V (N, V (N, V (N,.... It is easy to see that for any given N, U (N V (N are all rom vectors. Base on the exponential or Poisson assumptions of the arrival, service vacation processes, {( U (N, V (N, t } is an infinite-imensional Marov process whose state space E N is given by E N {(,, 3,...;,,,... : 3, 3, N Note that M (N l N l are nonnegative integers for l M (N l for l t, it is obvious that. Similarly, the fact that L (N L (N V (N V (N V (N for t can yiel that U (N U (N U (N.Furthermore, since the two rom variables U (N V (N l tae values in the set {, /N, /N,...,(N /N, } for, l t, this gives that for t, there exist two positive integers K L such that U (N U (N U (N (N K >, U for K ; }.
V (N V (N V (N L >, V(N l for l L. To analyze the infinite-imensional Marov process {( U (N, V (N, t } on state space E N,wewrite E U (N,, Let l E V (N l, l. (,, 3,... (,,,.... It is easy to see that with 3. In the remainer of this section, we set up an infinite-imensional system of ifferential equations whose purpose is to be able to etermine the expecte fraction vector (,.. The system of ifferential equations To etermine the expecte fraction vector (,,thissubsectionprovies aunifieeffectivemethotosetupaninfinite-imensionalsystemofifferential equations satisfie by the expecte fraction vector in terms of a technique of taile equations. To that en, we first provie an example with to inicate how to erive these ifferential equations. In the supermaret moel of N ientical servers, we nee to etermine the expecte change in the number of servers with at least customers over a small time perio, t, that is, we shall compute the rate that any arriving customer selects servers from the N servers inepenently uniformly at rom joins the one whose queue length is the shortest. From Figs. 3, itisseenthatanyarriving customer joins either server wors or server vacations among the selecte servers, thus we nee to consier the following two cases: Case one: Entering one woring server. Inthiscase,theratethatanyarriving customer joins a woring server with the queue length the
Vacation Vacation Vacation Vacation 3 Woring Woring Woring 3 Service process Arrival process Vacation process Fig. The state transition relation in the M/M/ queue with server vacations m woring servers one with the shortest queue m vacation servers one with the shortest queue Woring server area Vacation server area Vacation server area Woring server area -m vacation servers (a -m woring servers (b Fig. 3 Two ifferent cases when joining a woring server or a vacation server
queue lengths of the other selecte servers are not shorter than is given by where Nλ W (N (u, u ; v,v ; t u(n W (u, u ; v,v ; t t, ( m m m C j m j m C m m m m C m r r m u(n m u(n m j m m Cm m v(n m m j u(n r m r. m m It is necessary to provie a etaile interpretation for how to erive Eq.. Fromthejoiningprocessexpresseby(ainFig.3 from the set ecomposition of all possible events inicate in Fig. 4, itisseen that the probability W (N (u, u ; v,v ; t given in Eq. contain the following three parts. Part I: Neither of the selecte servers is taing a vacation, that is, each of the selecte servers is woring for service. In this case, the probability that any arriving customer joins a woring server with the queue length the queue Fig. 4 Set ecomposition of all possible events when joining a woring server Each of the selecte servers is woring for service, there is at least one woring server with the shortest queue length -. (Part I In the selecte servers, there is at least one woring server with the shortest queue length -, there exists at least one vacation server while the queue length of each vacation server is more than customers. (Part II In the selecte servers, there are at least one woring server with the shortest queue length - at least one vacation server with the shortest queue length -. (Part III
Part II: lengths of the other selecte woring servers are not shorter than is given by m m m u(n u(n m m m u(n, where C m!/ m! ( m! is a binomial coefficient, m u(n is the probability that any arriving customer who can only choose one queue maes m inepenent selections uring the m selecte woring servers with the queue length at time t. For the selecte servers, there is at least one woring server with customers, there exist at least one vacation server while the queue length of each vacation server is more than customers. In this case, the probability that any arriving customer joins a woring server with the shortest queue length ; for the other selecte servers, the queue lengths of the selecte woring servers are not shorter than, thereexistatleastonevacationserverwhilethe queue length of each vacation server is more than customers, is given by C m m m C j m j m u(n u(n m C j m j m j C m m j m j j. m u(n Part III: For the selecte servers, there are at least one woring server with customers at least one vacation server with customers. In this case, if there are the selecte m servers with the shortest queue length where there are m woring servers m m vacation servers, then the probability that any arriving customer joins a woring server is equal to m /m. Therefore, the probability that any arriving customer joins a woring server with the queue length,
the queue lengths of the other selecte servers are not shorter than, thereareatleastoneworingserver with customers at least one vacation server with customers, is given by m C m m m m C m r r m m m m Cm m u(n m C m r r m u(n r m m m Cm m m m v(n m r m u(n r m r. m m v(n Using the above three parts, Eq. can be obtaine immeiately. Besies the above analysis for the arrival process, in what follows we consier the service vacation processes. The rate that a customer leaves a server queue by customers is given by N t. ( The rate that a server queue by at least customers completes its vacation is given by Using Eqs., 3, weobtain E L (N Nλ N this gives t u(n λ by means of E Nθ t. (3 u(n W (N (u, u ; v,v ; t t t Nθ t, u(n W (N (u, u ; v,v ; t θ (4 L (N /N.
Case two: Entering one vacation server. In this case, the rate that any arriving customer joins a vacation server with the queue length the queue lengths of the other selecte servers are not shorter than is given by Nλ v(n V (N (u, u ; v,v ; t t, (5 where V (N (u, u ; v,v ; t m m m C j m j m C m m m m C m r r m v(n m v(n m j m m Cm m u(n m m j v(n r m r. m m Note that Eq. 5 can be erive similarly to that in Case one by means of (b in Figs. 3 5. Using a similar analysis to Eq. 4, it follows from Eq. 5 that t v(n λ v(n V (N (u, u ; v,v ; t θ. (6 (u ; v,v ; t, The following theorem simplifies expressions for V (N V (N (u, u ; v,v ; t W (N (u, u ; v,v ; t for. Note that the simplifie expressions will be a ey in our later stuy. Fig. 5 Set ecomposition of all possible events when joining a vacation server Each of the selecte servers is at vacation, there is at least one vacation server with the shortest queue length -. (Part I In the selecte servers, there is at least one vacation server with the shortest queue length -, there exists at least one woring server while the queue length of each woring server is more than customers. (Part II In the selecte servers, there are at least one vacation server with the shortest queue length - at least one woring server with the shortest queue length -. (Part III
Theorem V (N (u ; v,v ; t m m m, for W (N (u, u ; v,v ; t m u(n m m v(n V (N (u, u ; v,v ; t m v(n m. m u(n Hence, for we have W (N (u, u ; v,v ; t V (N (u, u ; v,v ; t. Proof It is easy to see that V (N (u ; v,v ; t m m m m C m r r m j j m m. For, weobtain W (N (u, u ; v,v ; t C m m u(n m u(n m u(n m
m C j m j m C m m m m C m r C r m m C j m j v(n m j m m Cm m m m r u(n m C m m m C C j j m m C j m j v(n j m u(n m r m u(n m j m m Cm m m C m m m C C j j v(n j m m m C m r r j m u(n m u(n m j m m Cm m j j m m m C m r r j r m u(n j r
m C m m m C m m m v(n m C m m m m m m m Cm m m m Cm m v(n m m m Cm m v(n m m m C m r r m u(n m u(n m m m C m r r m m m u(n m m m m m v(n similarly, we have V (N (u, u ; v,v ; t m m r r m C m m m u(n m u(n v(n, v(n m. m u(n This completes the proof. Set L (N (u ; v,v ; t V (N (u ; v,v ; t for L (N (u, u ; v,v ; t W (N (u, u ; v,v ; t V (N (u, u ; v,v ; t. The sequence: L (N (u ; v,v ; t L (N (u, u ; v,v ; t for, is calle the invariance of environment factor, which will play a ey role in our later stuy with respect to how to set up the system of ifferential equations.
Using some similar analysis to Eqs. 4 6,weobtainaninfinite-imensionalsystem of ifferential equations satisfie by the expecte fraction vector (, as follows: t v(n θ, (7 for t v(n t u(n λ t v(n λ L (N (u ; v,v ; t θ, (8 λ u(n with the bounary conition v(n L (N L (N (u, u ; v,v ; t θ (9 (u, u ; v,v ; t θ (, t, ( with the initial conitions { u (N ( g,, where l ( h l, l. g g g 3, h h h, h g. ( Remar If, thenw (N (u, u ; v,v ; t for V (N (u l, u l ; v l,v l ; t for l. Inthiscase,wehave λ v(n t v(n { λ θ. l l u(n l u(n } λ u(n l v(n v(n v(n l θ u(n Therefore, the system of ifferential equations (7 to( is the same as those in Vveensaya Suhov (5..3 A useful probabilistic interpretation In this subsection, we provie a useful probabilistic interpretation for the invariance of environment factor L (N (u ; v,v ; t L (N (u, u ; v,v ; t for, this will help us to further unerst the system of ifferential equations (7to(.
Using Theorem, it is easy to chec that V (N (u ; v,v ; t, for v(n L (N (u, u ; v,v ; t { v(n u(n L (N (u, u ; v,v ; t { v(n v(n L (N (u, u ; v,v ; t v,v ; t for, we introuce some notation v(n v(n } u(n v(n } To give the probabilistic interpretation for u(n. u(n u(n V (N (u ; v,v ; t, L (N (u, u ; W V are the events in which any arriving customer is reirecte to a woring server or a vacation server with the queue length,respectively. X enotes the number of times in the romize loa balancing policy we choose a server with the exactly queue length, where we o not istinguish woring vacation servers. Y enotes the number of times in the romize loa balancing policy we choose a server with whose queue length is not shorter than. Now, we compute the two probabilities P {W } P {V } for. Using the law of total probability, we obtain P {W } P {W X m, Y m} P {X m, Y m}. m We can compute the conitional probability P {W X m, Y m} u(n v(n u(n, (3 which is inepenent of the number m. Infact,theconitionalprobabilityiseasyto compute, e.g., by thining in terms of an urn moel with blac white balls, from which one raws m balls, blac ones with probability q u(n v(n u(n
white ones with probability q. Then, once m balls are extrace, one raws at rom one ball from the m ones. The probability of having chosen a blac ball is equal to q. By means of Mitzenmacher (996, we obtain P {X m, Y m} m v(n Base on Eqs. 3 4,wehavethefollowingprobabilisticsetting u(n L (N (u, u ; v,v ; t P {W }. Similarly, we have Thus we obtain v(n L (N (u, u ; v,v ; t P {V }. P {W } P {W X m, Y m} P {V } P {V X m, Y m}. (4 P {X m, Y m} m P {X m, Y m}. m 3Amean-fiellimit In this section, we use the operator semigroup to provie a mean-fiel limit for the sequence {( U (N, V (N, t } of infinite-imensional Marov processes for N,, 3,...,showthatthissequenceofMarovprocessesasymptotically approaches a single trajectory ientifie by the unique global solution to the infinite-imensional system of limiting ifferential equations. For the two vectors we write N (,, 3,... { (, : 3, (,,,...,, N N l are nonnegative integers for l } N {(, : (, N e e < }, where e is a column vector of ones with a suitable imension in the context. For the two vectors u (u, u, u 3,... v (v,v,v,v 3,...,set {(u, v : u u u 3, v v v v 3 }
{ (u, v : (u, v ue ve < }. Obviously, N N N. In the vector space, wetaeametric ρ ( (u, v, (u, v { { u u sup max, v v }} (5 for (u, v, (u, v. Note that uner the metric ρ ( (u, v, (u, v, the vector space is separable compact. For (g, h N,wewrite for L (g ; h, h L (g, g ; h, h C m (h h m (h g m, m C m (g g h h m (g h m. m Now, we consier the infinite-imensional Marov process { ( U (N, V (N, t } on state space N (or N in a similar analysis for N,, 3,... Note that the stochastic evolution of this supermaret moel of N ientical servers is escribe as the Marov process {( U (N, V (N, t },where ( U (N, V (N A N f ( U (N, V (N, t where A N acting on functions f : N R is the generating operator of the Marov process {( U (N, V (N, t }, A N A In N AOut N, (6 for (g, h N A In N f (g, h λn (g g L (g, g ; h, h f (g e N, h f (g, h λn (h h L (g ; h, h f (g, h e N f (g, h λn (h h L (g, g ; h, h f (g, h e N f (g, h (7
A Out N N θ N (g g f (g e N, h f (g, h h f (g e N, h e N f (g, h, (8 where e sts for a row vector with the th entry all others. For (g, h N, it follows from Eqs. 6 to 8 that A N f (g, h λn (g g L (g, g ; h, h f (g e N, h f (g, h λn (h h L (g ; h, h f (g, h e N f (g, h λn (h h L (g, g ; h, h f (g, h e N f (g, h N θ N (g g f (g e N, h f (g, h h f (g e N, h e N f (g, h The operator semigroup of the Marov process {( U (N, V (N, t } is efine as T N, whereif f : N C,thenfor ( g, h N t (9 T N f (g, h E f (U N, V N U N ( g, V N ( h. ( Note that A N is the generating operator of the operator semigroup T N, itiseasy to see that T N exp {A N t} for t. Definition Aoperatorsemigroup{S : t } on the Banach space L C( is sai to be strongly continuous if lim t S f f for every f L; itissaitobea contractive semigroup if S for t. Let L C( be the Banach space of continuous functions f : R with uniform metric f max f (u, similarly, let L N C( N. The inclusion u N inuces a contraction mapping N : L L N, N f (u f (u for f L u N. Now, we consier the limiting behavior of the sequence {(U (N, V (N, t } of Marov processes for N,, 3,... Two formal limits for the sequence {A N } of generating operators for the sequence {T N } of semigroups are expresse as A lim N A N T lim N T N for t, respectively. It follows from
Eq. 9 that as N A f (g, h λ We write for (g g L (g, g ; h, h g f (g, h λ (h h L (g ; h, h f (g, h h λn (h h L (g, g ; h, h f (g, h h (g g f (g, h θ g L (u ; v,v ; t h f (g, h g C m v v m v u m, m L (u, u ; v,v ; t m f (g, h. ( h u u v v m u v m. Let u lim N v lim N for t, where u lim N for v l lim N l for l. Baseonthelimiting generating operator A given in Eq., asn it follows from the system of ifferential equations (7 to( that(u, v is a solution to the following system of ifferential equations t v u u θv, ( for t v λ v v L (u ; v,v ; t θv, (3 t v λ v v L (u, u ; v,v ; t θv, (4 t u λ u u L (u, u ; v,v ; t u u θv, with the bounary conition (5 v u, t, (6
with the initial conitions { u ( g,, v l ( h l, l. (7 Remar In the next section, we shall prove that the vector (u(t, g, h, v(t, g, h is the unique global solution to the system of ifferential equations (to(7for t,whereu(, g, h g v(, g, h h. We efine a mapping: ( g, h (u(t, g, h, v(t, g, h, where ( u(t, g, h, v(t, g, h is a solution to the system of ifferential equations ( to(7. For the operator semigroup T acts in the space L. If f L (g, h, then T f (g, h f (u(t, g, h, v(t, g, h. (8 From Eqs. 9, it is easy to see that the operator semigroups T N T are strongly continuous contractive, see, for example, Section. in Chapter one of Ethier Kurtz (986. We enote by D(A the omain of the generating operator A. ItfollowsfromEq.8 that if f is a function from L has the partial erivatives g i f (g, h for i f (g, h L for j, sup i, j h j { } g i f (g, h, h j f (g, h <, then f D(A. Let D be the set of all functions f L that have the partial erivatives g i g j f (g, h, such that f (g, h, { sup i,i g i g i j, j (g,h g i h j f (g, h h j g i f (g, h, h i h j f (g, h, there exists C C( f < { } sup f (g, h i, j g, f (g, h i h < C (9 j f (g, h, f (g, h g i h, } f (g, h j h j h < C. (3 j We call that f L epens only on the first K two imensional variables if for (g (, h (, (g (, h (, it follows from g ( i g ( i for i K h ( j h ( j for j K that f (g (, h ( f (g (, h (.Asimilarsimpleprooftothatin Proposition in Vveensaya et al. (996 can show that the set of functions from L that epens on the first finite two imensional variables is ense in L. The following lemma comes from Proposition in Vveensaya et al. (996. We restate it here for convenience of escription. Lemma Consier an inf inite-imensional system of if ferential equations: For, z ( c z t z i a i, b, i
let ai, a, b b exp {bt}, c ϱ, b a < b. Then i z ϱ exp {at} b exp {bt} exp {at}. b a Definition Let A be a close linear operator on the Banach space L C(. A subspace D of D (A is sai to be a core for A if the closure of the restriction of A to D is equal to A,i.e.,A D A. We introuce some notations M 3 4 M M C m m m (, m C m ( m m m m ( (, C m (m m m m C m ( m( m m m 6 m 6 C m ( m(m m m m m (m ( m C m (m (m m 3 m m ( m( m, a M θ M a M θ M. The following lemma is a ey to prove that the set D is a core for the generating operator A. Lemma 3 Let (u, v be a solution to the system of if ferential equations ( to (7. Then { } u (t, g, h g, u (t, g, h i h exp {(M θ M t}, (3 j sup i, j sup i, j { } v (t, g, h g, v (t, g, h i h exp {(M θ M t}, (3 j
{ u (t, g, h sup g i g, u (t, g, h i g i h, } u (t, g, h j h j h j i,i j, j M M 3 exp {at} exp {at} M θ M (33 sup i,i j, j { v (t, g, h g i g, v (t, g, h i g i h, } v (t, g, h j h j h j M M 3 M θ M exp { a t } exp { a t }. (34 Proof We only prove (3, while (3to(34canbeprovesimilarly. It is easy to verify that the solution (u, v to the system of ifferential equations (to(7possessesthepartialerivatives u (t, g, h g j, u (t, g, h h j, v (t, g, h g j, v (t, g, h h j, u (t, g, h g i g j, u (t, g, h g i h j, u (t, g, h h i h j, v (t, g, h g i g j, v (t, g, h g i h j, v (t, g, h h i h j. In what follows we only compute the two erivatives u(t,g,h g j v(t,g,h h j,while the other erivatives can be compute similarly. For simplicity of escription, we write that u u (t, g, h, u, j u(t,g,h v (t,g,h h j. It follows from Eqs., 5 6 that for all, j, t u, j u, j u, j θv, j g j or v, j u, j t ( ( λ u, j u, j L (u, u ; v,v u, j u, j θv, j λ (u u L, j (u, u ; v,v,
L, j (u, u ; v,v ; t C m ( m(u u v v m m (v u m ( v, j u, j C m (m (u u v v m m ( (v u m u, j u, j v, j v, j. Using Lemma, we obtain Inequalities (3with a M θ M, a M θ M, b, b, ϱ. This completes the proof. Lemma 4 The set D is a core for the generating operator A. Proof It is obvious that D is ense in L D D(A. Let D be the set of functions from D, which epen only on the first finite two imensional variables. It is easy to see that D is ense in L. Using Proposition 3.3 in Chapter of Ethier Kurtz (986, it can show that for any t, the operator semigroup T oes not bring D out of D. Select an arbitrary function ϕ D let f (g, h ϕ(u(t, g, h, v(t, g, h for (g, h. ItfollowsfromLemma3that f has the partial erivatives u (t, g, h g j, u (t, g, h h j, v (t, g, h g j, v (t, g, h h j, u (t, g, h g i g j, u (t, g, h g i h j, u (t, g, h h i h j, v (t, g, h g i g j, v (t, g, h g i h j, v (t, g, h h i h j. they satisfy the inequalities (3 to(34. Therefore f D. This completes the proof. The following theorem applies the operator semigroup to provie an mean-fiel limit, which shows that the sequence {( U (N, V (N, t } of Marov processes asymptotically approaches a single trajectory ientifie by the unique global solution to the infinite-imensional system of ifferential equations (to(7. Theorem For any continuous function f : R t >, lim sup TN f (g, h f (u(t, g, h, v(t; g, h, (g,h N N the convergence is uniform in t within any boune interval.
Proof This proof is to use the convergence of the operator semigroups as well as the convergence of their corresponing generating generators, e.g., see Theorem 6. in Chapter of Ethier Kurtz (986. Lemma 4 shows that the set D is a core for the generating operator A. For any function f D, wehave N f (g e N, h f (g, h f (g, h g N f (g, h e N f (g, h f (g, h h γ, (g f (g γ, (g e N where < γ i, (g, γ i, (h < for i,. Since γ, (g f (g γ, (g e N, h N C N g g N, h, γ, (h f (g, h γ, (h e N g N, γ, (h f (g, h γ, (h e N g N C N, we obtain A N f (g, h f (g, h C (g g L (g, g ; h, h N (h h L (g, g ; h, h (h h L (g ; h, h (g g θ h { C } M (g l g l (h h g θ h N l C M (g h g θ h. N Note that C, M h are all finite, it is clear that as N, lim sup A N f (g, h A f (g, h. (g,h N N This completes the proof. Remar 3 ( As iscusse in Ethier Kurtz (986, there have been at least three basic techniques: operator semigroup, martingale ensity epenent population process, for analyzing the wea approximation of the sequences of Marov processes. In fact, the three techniques have been applie to the mean-fiel limit
of supermaret moels up to now, e.g., see the operator semigroup by Vveensaya et al. (996, the ensity epenent population process by Mitzenmacher (996 the martingale by Turner (996, 998. In this section, we use the operator semigroup to provie a mean-fiel limit for the supermaret moel with server multiple vacations, show that the sequence of corresponing Marov processes asymptotically approaches a single trajectory ientifie by the unique global solution to the infinite-imensional system of mean-fiel limit equations. ( The mean-fiel limit over the finite time intervals has been generalize in the PhD thesis of Mitzenmacher (996 to the infinite-imensional case when the right-h sie of the system of ifferential equations is Lipschitz. Therefore, using the generalize results by Mitzenmacher (996, we may significantly simplify some results in this section, also we can possibly obtain a stronger form of convergence some error bouns. Furthermore, reaers may refer to Graham (a, b, 4 Lucza McDiarmi (6, 7 forthe longest queue length, the asymptotic inepenence chaoticity on path space. 4Existenceuniqueness In this section, we first provie an effective technique to organize the Lipschitzian conition of the infinite-imensional fraction vector function F : R C (R. Note that the Lipschitzian conition for 3 is always ifficult in the literature. Then we apply the Lipschitzian conition, together with the Picar approximation, to show that the limiting expecte fraction vector is the unique global solution to the system of ifferential equations. For convenience of escription, we write that u u (t, g, h for v l v l (t, g, h for l. Usingv (t, g, h u (t, g, h, the system of ifferential equations (to(7 can be simplifie as an initial value problem as follows for, t v u u θv, (35 t v λ (v v L (u ; v,v ; t θv, (36 t v λ (v v L (u, u ; v,v ; t θv (37 t u λ (u u L (u, u ; v,v ; t (u u θv, (38 with the bounary conition with the initial conition v u (39 { u ( g,, v l ( h l, l. (4
4. A Lipschitzian conition To establish the Lipschitzian conition, we nee to use the erivative of the infiniteimensional function G : R C (R. Thus it is necessary to provie some useful notation efinitions of erivatives as follows. For the infinite-imensional function G : R C (R,wewritex (x, x, x 3,... G(x (G (x, G (x, G 3 (x,..., wherex G (x are scalar for. Then the matrix of partial erivatives of the infinite-imensional function G(x is efine as G (x x G (x DG(x x G (x x 3. G (x x G (x x G (x x 3. G 3 (x x G 3 (x x G 3 (x. (4 x 3. Now, we efine two classes of erivatives for the infinite-imensional function G(x. In fact, they are a irect minor generalization from the erivatives of finiteimensional functions, e.g., see Chapter of Taylor (996 Chapter 3 of Fleming (977 for more etails. Definition 3 Let the infinite-imensional function G : R C (R. ( If there exists a linear operator A : R R a scalar t R such that for any vector h R G (x ht G (x hat lim, t t then the function G (x is calle to be Gateaux ifferentiable at x R.Inthis case, we write the Gateaux erivative G G (x A. ( If there exists a linear operator B : R R such that for any vector h R G (x h G (x hb lim, h h then the function G (x is calle to be Frechet ifferentiable at x R.Inthis case, we write the Frechet erivative G F (x B. It is easy to chec that if the infinite-imensional function G (x is Frechet ifferentiable, then it is also Gateaux ifferentiable. At the same time, we have G G (x G F (x DG(x. (4
Let t (t, t, t 3,... with t for. Then we write DG(x t (y x G (x t (y x x G (x t (y x x G (x t (y x x 3. G (x t (y x x G (x t (y x x G (x t (y x x 3. G 3 (x t 3 (y x x G 3 (x t 3 (y x x G 3 (x t 3 (y x. x 3. The following lemma provies two useful properties for the Gateaux erivatives of the infinite-imensional functions. Obviously, the two useful properties also hol for the Frechet erivatives. Lemma 5 If the inf inite-imensional function G : R C (R is Gateaux ifferentiable, then there exists a vector t (t, t, t 3,... with t for such that Furthermore, we have G (y G (x (y x DG(x t (y x. (43 G (y G (x sup DG(x t (y x y x. (44 t Proof For the function G (x,itiseasytochecthatthereexistsanumbert, such that G (y G (x Note that we obtain Since it follows i (y i x i G (x t (y x x i ( G (x t (y x (y x, G (x t (y x T,.... x x G (y G (x (G (y G (x, G (y G (x, G 3 (y G 3 (x,..., G (y G (x (y x DG(x t (y x. DG(x t (y x sup DG(x t (y x, t G (y G (x sup DG(x t (y x y x. t This completes the proof.
Note that v u, wesetthatx (u,v ; u,v ; u 3,v 3 ; u 4,v 4 ;... F(x (F (x, F (x, F 3 (x,..., where F (x (u u θv, λ(v v L (u ; v,v ; t θv (45 F (x (λ(u u L (u, u ; v,v ; t (u u θv, λ(v v L (u, u ; v,v ; t θv. (46 Then F (x is in C ( R, the system of ifferential vector equations (35to(4 is rewritten as with initial conition x F (x (47 t x ( (g, h ; g, h ; g 3, h 3 ; g 4, h 4 ;.... (48 In what follows we show that the infinite-imensional function F(x is Lipschitzian for t. From ( of Definition 3 Eq. 4, the matrix of partial erivatives of the function F(x is given by A (x B (x C (x A (x B 3 (x DF(x C (x A 3 (x B 4 (x, (49......... where A (x is a matrix of size for, thesizesofalltheotherscanbe etermine accoringly. Let for Then for A (x L L (u ; v,v ; t V (u ; v,v ; t L W (u, u ; v,v ; t V (u, u ; v,v ; t. A (x θ λ (v v L u λl λ (v v L v, θ λl λ (u u L u λ (v v L u λ (u u L v θ λv λ (v v L v, θ B (x λl λ (u u L u λ (u u L v λ (v v L u λv λ (v v L v,
C j (x (, j, where L u C m ( m v v m v u m, m for L u L v C m ( m v v m v u m m C m ( m v v m v u m, m C m ( m u u v v m u v m m C m ( m u u v v m u v m, m L v C m ( m u u v v m u v m m C m ( m u u v v m u v m, m L u C m (m u u v v m u v m, m L v C m (m u u v v m u v m. m Hence it follows from Eq. 49 that { e T DF(x max A (x B (x, sup e T C (x A (x B (x }. (5 Let M 5 m ( m.
Base on the expression L (u ; v,v ; t L (u, u ; v,v ; t for, itis easy to chec that for L (u ; v,v ; t M L (u, u ; v,v ; t M. At the same time, we have for L u L u L v L v L u L v M 5, M 5, M 5. Note that e T A (x C (x ( L θ, λv λ (v v L θ, u v we obtain e T A (x B (x λm λm 5 θ. (5 Since for e T A (x B (x C (x ( L λ (u u u L u L v L θ, v L λ (v v L L L u u v v θ, we obtain e T A (x B (x C (x 4λM5 θ. (5 Then it follows form Eqs. 5, 5 5 that DF(x M, where M max (λm λm 5 θ, 4λM 5 θ.
Note that for (u, v, (ũ, ṽ, itfollowsfromeq.43 that F(u, v F(ũ, ṽ (u, v (ũ, ṽ DF((u, v t (u, v (ũ, ṽ from Eq. 44 that F(u, v F(ũ, ṽ sup DF((ũ, ṽ t (u, v (ũ, ṽ (u, v (ũ, ṽ t M (u, v (ũ, ṽ. (53 Therefore, the function F(u, v is Lipschitzian for (u, v. Remar 4 Let G : R m R n be continuously ifferentiable. Then Proposition 4.5 or Proposition 4.4 in Fleming (977 prove that the function G is locally Lipschitzian for t. Note that Eq. 53 extens the Lipschitzian conition to the infinite-imensional continuously ifferentiable function F : R R,sucha generalization is always necessary in the stuy of supermaret moels. Remar 5 The Lipschitzian conition for 3 is always ifficult in the literature, while few available results were organize in the supermaret moels with,e.g., see Vveensaya Suhov (997, 5Mitzenmacheretal.(. Therefore, here we provie a unique effective metho to compute the Lipschitzian conition for more complicate supermaret moels with. 4. The Picar approximation In this subsection, we apply the Lipschitzian conition, together with the Picar approximation, to show that the limiting expecte fraction vector is the unique global solution to the system of ifferential equations. Let v u x (u,v ; u,v ; u 3,v 3 ; u 4,v 4 ;...Wewrite {x : u u, v v v 3 } It follows from Eqs. 47 to 48 that for x this gives where x x ( t F(x (ss, x (g, h t F(x (ss, (54 ( g, h (g, h ; g, h ; g 3, h 3 ; g 4, h 4 ;.... Base on the integral equation (54, the following theorem inicates that there exists the unique global solution to the system of ifferential equations (35 to (4fort. Theorem 3 For (g, h, there exists the unique global solution to the Eq. 54 for t.
Proof We tae the Picar sequence as follows for n It follows from Eq. 53 that x, x n (g, h t F ( x n (s s. x n x n t F ( x n (s F ( x n (s s Mt x n x n (Mtn x x. (n! From the bounary conition: x x, itisclearthatif t /M, thenlim n (Mt n /(n!, whichleastothatasn,thepicar sequence {x n } is uniformly convergent for t, /M. Let x lim n x n for t, /M. Then x is a solution to Eq. 54 for t, /M. Let y is another solution to Eq. 54 for t, /M. Then it is easy to chec that this gives that for t, /M, x y (Mtn (n!, y lim n x n x. This shows that x is the unique solution to Eq. 54 for t, /M. We consier the following equation Tae the Picar sequence for n x x( t M F(x (ss. M x, x n x( t M F ( x n (s s. It is easy to show that x lim n x n is the unique solution to Eq. 54 for t /M, /M. We assume that for l, x is the unique solution to Eq. 54 for t /M,( /M. Then for l, we consier the following equation ( t x x M M M F(x (ss.
Tae the Picar sequence x for n x n x( t M M F(x n (ss. It is easy to inicate that x lim n x n is the unique solution to Eq. 54 for t ( /M,( /M. By inuction, it is easy to show that x lim n x n is the unique solution to Eq. 54 for t l/m,(l /M for l,,,... Note that,, M M, M M, 3, M thus, x is the unique global solution to Eq. 54 for t. This completes the proof. Remar 6 Comparing with the finite-imensional system of integral equations (e.g., see Chapter of Hale (98, Theorem 3 maes some necessary generalization of the Picar approximation in orer to eal with existence uniqueness of solution to the infinite-imensional system of integral equations. Note that such a generalization is always necessary in the stuy of supermaret moels. Remar 7 Note that the infinite-imensional system of ifferential equations for the supermaret moel with server multiple vacations is efine on a Banach space, the existence uniqueness of solution is immeiately obtaine ue to the existing results on Banach spaces given that the right-h sie of the infinite-imensional system of ifferential equations is Lipschitzian. 5 The fixe point In this section, we analyze the fixe point of the infinite-imensional system of ifferential equations ( to(7, set up an infinite-imensional system of nonlinear equations satisfie by the fixe point. Base on this, we provie an effective algorithm for computing the fixe point. Note that the fixe point is a ey in performance analysis of this supermaret moel. Let π (π (W, π (W, π (W 3,...; π (V, π (V, π (V, π (V 3,... The row vector π is calle a fixe point of the infinite-imensional system of ifferential equations ( to(7 if π lim t (u, v, whereπ (W lim t u for π (V l lim t v l for l.itiswell-nownthatifπ is a fixe point of the limiting expecte fraction vector (u, v, then lim t t u, lim t t v. (55
For two sequences of positive numbers {x : } {y l : l } with x y, we introuce two function notations for L (x ; y, y C m (y y m (x y m, m L (x, x ; y, y C m (x x y y m (x y m. m Taing t in both sies of the system of ifferential equations ( to(7, we obtain π (W π (W θπ (V, (56 ( λ π (V π (V L π (W ; π (V, π (V θπ (V, (57 for, ( λ π (W π (W L π (W, π (W ; π (V, π (V π (W π (W θπ (V (58 ( λ π (V π (V L π (W, π (W ; π (V, π (V θπ (V, (59 with the bounary conition π (V π (W. (6 To solve the system of nonlinear equations, it is a ey to first compute the bounary probabilities π (V π (W in the fixe point. The following theorem provie an effective metho to etermine the bounary probabilities. Theorem 4 If λ <,thenthebounaryprobabilitiesπ (V π (W are given by Proof For,wehave ( π (V π (V L π (W ; π (V, π (V π (V λ π (W λ. m π (V π (W m m π (V π (V π (V π (W π (V π (W,
( π (V π (V L π (W, π (W ; π (V, π (V π (V π (V m π (V π (V π (V π (W π (W π (V π (V π (V π (V { π (V π (W π (W π (W π (V m π (W m } π (W ( π (W π (W L π (W, π (W ; π (V, π (V π (W π (W π (V π (V π (W π (W { π (V π (W π (V } π (W, this gives ( π (V π (V L π (W ; π (V, π (V ( π (V π (V L π (W, π (W ; π (V, π (V ( π (W π (W L π (W, π (W ; π (V, π (V π (V π (W π (V π (W π (V π (W, { π (V π (W π (V Together with π (V π (W, it follows from Eqs. 56 to 6 that { ( π (W λ π (W π (W L π (W, π (W ; π (V, π (V π (V π (V L ( Thus, using π (V π (W we obtain π (W ; π (V, π (V } π (W ( } π (V π (V L π (W, π (W ; π (V, π (V λ. (6 π (V λ. This completes the proof.
Remar 8 Note that ( π (W π (W L π (W, π (W ; π (V, π (V ( π (V π (V L π (W, π (W ; π (V, π (V, ( π (V π (V L π (W ; π (V, π (V the sequence { ( ( π (W π (W L π (W, π (W ; π (V, π (V ; π (V π (V L π (W ; π (V, π (V, ( } π (V π (V L π (W, π (W ; π (V, π (V : is a probability vector. Now, we provie an iterative algorithm for computing the fixe point of the infinite-imensional system of ifferential equations (56to(6. We efine F (x θ x λ (ξ x L (δ, δ ; ξ, x,. (6 We assume that ξ λ δ λ. UsingEq.6, wetaethatπ (V ξ δ. We enote by ξ asolutionin(, ξ to the nonlinear equation π (W set F (x θ x λ (ξ x L (δ ; ξ, x, (63 Let ξ be a solution in (, ξ to the nonlinear equation set δ δ θξ. (64 F (x θ x λ (ξ x L (δ, δ ; ξ, x, (65 δ 3 δ θξ λ (δ δ L (δ, δ ; ξ, ξ. (66 We assume that the pairs (ξ, δ, (ξ, δ,...,(ξ, δ have been obtaine iteratively. Then we enote by ξ asolutionin(, ξ to the nonlinear equation F (x, set δ δ θξ λ (δ δ L (δ, δ ; ξ, ξ. (67 It is clear that < ξ < ξ < < ξ < ξ λ < δ < δ < < δ < δ λ. The following theorem expresses the fixe point of the infinite-imensional system of ifferential equations (56to(6by means of the iterative algorithm.note that it is a ey in the proof that we nee to inicate the uniqueness of the sequences {(ξ, δ : }.
Theorem 5 If λ <, then the fixe point π (π (W, π (W, π (W 3,...; π (V, π (V, π (V,...is uniquely given by π (V l ξ l, l, π (W δ,. Proof It is obvious that π (V ξ λ π (W δ λ. It follows from Eq. 57 that ( λ π (V π (V L π (W ; π (V, π (V θπ (V, that is Let λ m m ( m λ π (V π (V λ θπ (V. F (x θ x λ C m λ xm (x λ m m θ x λ λ x λ. Then F ( λ λ <, F (ξ θξ > for x (, ξ x F (x θ λ x λ >. Note that F (x is a continuous function for x (, ξ, thus there exists a unique positive solution x ξ to the nonlinear equation F (x for x (, ξ. Hence, π (V ξ.itfollowsfromeq.56 that π (W π (W θπ (V δ θξ δ. It follows from Eq. 59 for that ( λ π (V π (V L π (W, π (W ; π (V, π (V θπ (V, that is λ ξ π (V L ( δ, δ ; ξ, π (V θπ (V. Let F (x θ x λ ξ x L (δ, δ ; ξ, x ξ x θ x λ (ξ δ (x δ. ξ x δ δ
Then F ( λξ L (δ, δ ; ξ, <, F (ξ θξ > for x (, ξ ( ( x F (x θ λ (ξ x(ξ δ λ (ξ x(x δ x ξ x δ δ x ξ x δ δ θ λ (ξ δ (ξ x δ δ (ξ x δ δ λ (ξ x(ξ δ (ξ x δ δ λ (x δ (ξ x δ δ (ξ x δ δ λ (ξ x(x δ (ξ x δ δ λ (ξ x(x δ (ξ x δ δ (ξ x δ δ >. Since F (x is a continuous function for x (, ξ, there exists a unique positive solution x ξ to the nonlinear equation F (x for x (, ξ.hence,π (V ξ. It follows from Eqs. 58 67 that ( π (W 3 π (W θπ (V λ π (W π (W L π (W, π (W ; π (V, π (V δ θξ λ δ δ L (δ, δ ; ξ, ξ δ 3. We assume that for l, π (V ξ π (W δ, where < ξ < ξ < < ξ < ξ λ < δ < δ < < δ < δ λ. Then for l, itfollows from Eq. 59 that ( λ π (V π (V L π (W, π (W ; π (V, π (V θπ (V, that is λ ξ π (V L ( δ, δ ; ξ, π (V θπ (V. Let F (x θ x λ ξ x L (δ, δ ; ξ, x ξ x θ x λ (ξ δ (x δ. ξ x δ δ Then F ( λξ L (δ, δ ; ξ, <, F (ξ θξ > for x (, ξ ( ( x F (x θ λ (ξ x(ξ δ λ (ξ x(x δ x ξ x δ δ x ξ x δ δ θ λ (ξ δ (ξ x δ δ (ξ x δ δ λ (ξ x(ξ δ (ξ x δ δ λ (x δ (ξ x δ δ (ξ x δ δ λ (ξ x(x δ (ξ x δ δ λ (ξ x(x δ (ξ x δ δ (ξ x δ δ >.
Note that F (x is a continuous function for x (, ξ,thereexistsaunique positive solution x ξ to the nonlinear equation F (x for x (, ξ. Hence, π (V ξ. It follows from Eqs. 58 67 that ( π (W π (W θπ(v λ π (W π (W L π (W, π (W ; π (V, π (V δ θξ λ δ δ L (δ, δ ; ξ, ξ δ. By inuction, this completes the proof. Uner the conition λ <, the following theorem escribes two important limiting processes which are relate to the fixe point. The two limiting processes are interesting when the convergence of fraction vector sequence is unerstoo as N /or t. Here, we omit its proof, while the proof can be complete by a similar iscussion to those of Theorem (iii Theorem 4 in Martin Suhov (999. Theorem 6 ( If λ <,thenforany(g, h ( u(t, g, h, v(t, g, h π. lim t Furthermore, there exists a unique probability measure ϕ on, which is invariant uner the map (g, h ( u(t, g, h, v(t, g, h, that is, for any continuous function f : R t > f (g, hϕ(g, h f (u(t, g, h, v(t, g, hϕ(g, h. Also, ϕ δ π is the probability measure concentrate at the f ixe point π. ( If λ <, thenforafixenumbern,, 3,...,theMarovprocess { (U (N, V (N, t } is positive recurrent, hence it has a unique invariant istribution ϕ N.Furthermore,{ϕ N } wealy converges to δ π,thatis,foranycontinuous function f : R lim E ϕ N f (g, h f (π. N Base on Theorems 6, we obtain a useful relation as follows lim lim ( (t, g, h, (t, g, h lim lim ( (t, g, h, (t, g, h π. t N Therefore, we have N t ( lim (t, g, h, (t, g, h π. N t 6 Performance analysis numerical examples In this section, we provie performance analysis of the supermaret moel with server multiple vacations, incluing the mean of the stationary queue length in any
server the expecte sojourn time that any arriving customer spens in this system. Furthermore, we use some numerical examples to analyze how the two performance measures epen on some crucial factors of this supermaret moel. Let Q be the stationary queue length of any server. Then Note that where E Q P {Q } π (W π (V (ξ δ. (68 E Q E Q this server is at vacation E Q, this server is woring, E Q, this server is at vacation E Q, this server is woring π (V π (W ξ δ. If λ <, then this supermaret moel with server multiple vacations is stable. In this case, we enote by S the sojourn time that any arriving customer spens in this system. It is easy to see that E S, this server is at vacation ( θ (ξ ξ L (δ ; ξ ; ξ E S, this server is woring thus we obtain ( θ (ξ ξ L (δ, δ ; ξ ; ξ (δ δ L (δ, δ ; ξ ; ξ, E S E S, this server is at vacation E S, this server is woring ( θ (ξ ξ L (δ ; ξ ; ξ θ ( θ(ξ ξ L (δ, δ ; ξ ; ξ θ (δ δ L (δ, δ ; ξ ; ξ. (69 In the remainer of this section, using Eqs. 68 69 we provie some numerical examples to analyze how the two performance measures E Q E S epen on some crucial parameters of this supermaret moel uner several ifferent values of.
Fig. 6 The stationary expecte queue length epens on vacation or no vacation 8 6 4,No Vacation,Vacation,No Vacation,Vacation θ,λ.5:.:.95 EQ 8 6 4.5.5.5.35.45.55.65.75.85.95 λ ( The role of vacation processes In this supermaret moel, we assume that the exponential service rate µ,the exponential vacation rate θ the Poisson arrival rate λ (.5,.95. Figures 6 7 inicate how the two performance measures of the supermaret moel epen on the role playe by the vacation processes, where λ (.5,.95. Figure 6 shows that the vacation processes increase E Q uner the two cases with,. At the same time, the choice number ecreases E Q when the servers have either vacations or no vacations. Fig. 7 The expecte sojourn time epens on vacation or no vacation 5,No Vacation,Vacation,No Vacation,Vacation θ,λ.5:.:.95 5 ES 5.5.5.5.35.45.55.65.75.85.95 λ
Fig. 8 The mean of the stationary queue length vs λ for the ifferent values of 9 8 7 3 5 θ,λ.5:.:.95 6 5 EQ 4 3.5.5.35.45.55.65.75.85 λ Figure 7 illustrates that the vacation processes also increase E S uner the two cases with,. Whentheservershaveeithervacationsornovacations,E S ecreases as the choice number increases. ( The role of Poisson arrival rates In this supermaret moel, we assume that the exponential service rate µ,the exponential vacation rate θ.4 the Poissonarrivalrate λ (.5,.95. Figures 8 9 inicate how the two performance measures of the supermaret moel epen on the Poisson arrival rate λ (.5,.95. Fig. 9 The expecte sojourn time vs λ for the ifferent values of 9 8 3 5 θ,λ.5:.:.95 7 ES 6 5 4 3.5.5.5.35.45.55.65.75.85.95 λ
Fig. The mean of the stationary queue length vs θ for the ifferent values of 3.5 3 λ.5,θ.:.: 3 5.5 EQ.5.5..4.6.8..4.6.8 θ Figure 8 shows that E Q increases as λ increases uner the three cases with,, 5. However,theroleofchoicenumber is complicate; only when λ is big, E Q ecreases as increases. Figure 9 illustrates that E S increases as λ increases uner the four cases with,, 3, 4. Atthesametime,E S ecreases as the choice number increases, this is ifferent from that in E Q. (3 The role of exponential vacation rates In this supermaret moel, we assume that the exponential service rate µ,the Poisson arrival rate λ.5 the exponential vacation rate θ (.,. Fig. The expecte sojourn time vs θ for the ifferent values of 7 6 λ.5,θ.:.: 3 5 5 ES 4 3..4.6.8..4.6.8 θ
Figures inicate how the two performance measures of the supermaret moel epen on the exponential vacation rate θ (.,. Figure shows that E Q ecreases as θ increases uner the four cases with,, 3, 4. However,theroleofchoicenumber is complicate, we can not escribe how E Q epens on yet. Figure illustrates that E S ecreases as θ increases uner the four cases with,, 3, 4. Atthesametime,E S ecreases as the choice number increases, this is ifferent from that in E Q. 7Concluingremars In this paper, we first analyze a supermaret moel of N ientical servers with server multiple vacations, set up an infinite-imensional system of ifferential equations satisfie by the expecte fraction vectors in terms of the technique of taile equations. Then, as N we use the operator semigroup to provie a meanfiel limit for the sequence of Marov processes, which wealy converges to the unique global solution for the infinite-imensional system of limiting ifferential equations. Finally, we provie an effective algorithm for computing the fixe point of the infinite-imensional system of limiting ifferential equations. Using the fixe point, we provie performance analysis of this supermaret moel, also give some numerical examples to analyze how the two performance measures epen on some crucial factors of this supermaret moel. This paper provies a clear picture for how to use the mean-fiel moels to numerically analyze performance measures of complicate supermaret moels, this picture is organize into three ey parts: ( Setting up system of ifferential equations, ( strict proofs of the mean-fiel limit (3 performance analysis of system. Therefore, the metho given in this paper can be applie to performance analysis of complicate supermaret moels with more rom factors, such as cases where each server is a retrial queue or a processor-sharing queue, each server may be failure repaire, the customers may be impatient or negative. Along these lines, there are a number of interesting irections for potential future research, for example:. We nee to evelop effective algorithms for computing the fixe point of complicate supermaret moels, specifically, to analyze the influence of crucial rom factors on the esign of algorithms. In fact, this paper inicates that the nonlinear ynamics of the supermaret moel with server multiple vacations maes the system of limiting ifferential equations more complicate, maing the computation of the fixe points more ifficult challenging. On the other h, it is worthwhile to note that for a Marov process, the fixe point with nonlinear structure is very ifferent from the stationary probability vectors with linear structure (e.g., see the RG-factorizations given in Li (, therefore there are many interesting topics for computing the fixe point with the nonlinear ynamic structure.. How to apply the operator semigroup to provie the mean-fiel limit in the stuy of supermaret moels with either non-poisson inputs or with non-exponential service times is still an open interesting problem. Reaer may refer to, such
as, Bramson et al. (,,, Li Lui ( Li(. Recently, Li et al. ( maecrucialavancesinapplyingtheoperatorsemigroup the mean-fiel limit to analyzing the supermaret moel with PH service times. However, we believe that a large gap still exists for ealing with either renewal inputs or general service times, because of the fact that a more complicate nonlinear ynamic structure exists nees to be given a etaile analysis in orer to set up the infinite-imensional system of ifferential equations. Acnowlegements The authors than the two referees for their valuable comments remars, acnowlege Professor Benny van Hout for many valuable suggestions to sufficiently improve the system of ifferential equations erive in Section of this paper. At the same time, Q.L. Li Y. Wang than that this research is partly supporte by the National Natural Science Founation of China (No. 7787, No. 675 the Hebei Natural Science Founation of China (No. A35. References Alfa AS (3 Vacation moels in iscrete time. Queueing Syst 44:5 3 Benaim M, Le Bouec JY (8 A class of mean-fiel interaction moels for computer communication systems. Perform Eval 65:83 838 Borenave C, McDonal D, Proutiere A (9 A particle system in interaction with a rapily varying environment: mean-fiel limits applications. Available: arxiv:7363v3 Le Bouec JY, McDonal D, Muninger J (7 A generic mean-fiel convergence result for systems of interacting objects. In: Proc. Conf. IEEE on the quantitative evaluation of systems, pp 3 8 Bramson M (8 Stability of queueing networs. Springer-Verlag, New Yor Bramson M ( Stability of join the shortest queue networs. Ann Appl Probab :568 65 Bramson M, Lu Y, Prabhaar B ( Romize loa balancing with general service time istributions. In: Proceeings of the ACM SIGMETRICS international conference on measurement moeling of computer systems, pp 75 86 Bramson M, Lu Y, Prabhaar B ( Decay of tails at equilibrium for FIFO join the shortest queue networs. Available: arxiv:6.458 Bramson M, Lu Y, Prabhaar B ( Asymptotic inepenence of queues uner romize loa balancing. Queueing Syst 7:47 9 Doshi BT (986 Queueing system with vacation: a survey. Queueing Syst :9-66 Doshi BT (99 Generalization of the stochastic ecomposition results for single server queues with vacations. Stoch Moel 6:37 333 Dshalalow JH (995 Avances in queueing theory, methos open problems. CRC Press, Boca Raton Dshalalow JH (997 Frontiers in queueing moels applications in science engineering. CRC Press, Boca Raton Ethier SN, Kurtz TG (986 Marov processes: characterization convergence. John Wiley & Sons, New Yor Fleming W (977 Functions of several variables, n en. Springer, New Yor Foss S, Chernova N (998 On the stability of a partially accessible multi-station queue with stateepenent routing. Queueing Syst 9:55 73 Fuhrmann SW, Cooper RB (985 Stochastic ecomposition in the M/G/ queue with generalize vacation. Oper Res 33:7 9 Gast N, Gaujal B (9 A mean-fiel approach for optimization in particles systems applications. Technical Report N 6877, INRIA Rocquencourt, Frence Gast N, Gaujal B ( Marov chains with iscontinuous rift have ifferential inclusions limits. Technical Report N 735, INRIA Rocquencourt, Frence Gast N, Gaujal B, Le Bouec JY ( Mean-fiel for Marov ecision processes: from iscrete to continuous optimization. Technical Report N 739, INRIA Rocquencourt, Frence Graham C (a Kinetic limits for large communication networs. In: Bellomo N, Pulvirenti M (es Moelling in applie sciences. Birhäuser, pp 37 37
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Quan-Lin Li is Full Professor in School of Economics Management Sciences, Yanshan University, Qinhuangao, China. He receive the Ph.D. egree in Institute of Applie Mathematics, Chinese Acaemy of Sciences, Beijing, China in 998. He has publishe a boo (Constructive Computation in Stochastic Moels with Applications: The RG-Factorizations, Springer, over 4 research papers in a variety of journals, such as, Avances in Applie Probability, Queueing Systems, Stochastic Moels, European Journal of Operational Research, Computer Networs, Performance Evaluation, Discrete Event Dynamic Systems, Computers & Operations Research, Computers & Mathematics with Applications, Annals of Operations Research, International Journal of Prouction Economics. His main research interests concern with Queueing Theory, Stochastic Moels, Matrix-Analytic Methos, Manufacturing Systems, Computer Networs, Networ Security, Supply Chain Ris Management. Guirong Dai is a master stuent in School of Economics Management Sciences, Yanshan University, Qinhuangao, China. Her research interests inclue Queueing Theory, Stochastic Moels, Computer Networs.
John C. S. Lui (M93-SM-F was born in Hong Kong. He receive the Ph.D. egree in computer science from the University of California, Los Angeles, 99. He is currently a Professor with the Department of Computer Science Engineering, The Chinese University of Hong Kong (CUHK, Hong Kong. He was the chairman of the Department from 5 to. His current research interests are in communication networs, networ/system security (e.g., clou security, mobile security, etc., networ economics, networ sciences (e.g., online social networs, information spreaing, etc., clou computing, large-scale istribute systems, performance evaluation theory. Professor Lui is a Fellow of the Association for Computing Machinery (ACM, a Fellow of IEEE, a Croucher Senior Research Fellow, an electe member of the IFIP WG 7.3. He serves on the Eitorial Boar of IEEE/ACM Transactions on Networing, IEEE Transactions on Computers, IEEE Transactions on Parallel Distribute Systems, Journal of Performance Evaluation International Journal of Networ Security. He receive various epartmental teaching awars the CUHK Vice-Chancellors Exemplary Teaching Awar. He is also a co-recipient of the IFIP WG 7.3 Performance 5 IEEE/IFIP NOMS 6 Best Stuent Paper Awars. Yang Wang is an Assistant Professor in the Department of Computer Science Technology, Peing University, Beijing, PR China. She receive her Ph.D. egree from the Department of Computer Science Technology, Tsinghua University in 9. Her research interests inclue networ security, system performance evaluation, wireless networs communication networs etc.